Implicit function theorem in Physics-Informed Neural Networks to solve parameterized differential equations

Julien Marie-Anne, Cyriaque Rousselot, Nilo Schwencke, Alena Shilova

Keywords: pinn, curriculum learning, parameterized differential equation, natural gradient descent

TL;DR: A generalization to implicit function theorem for parameterized differential equations.

Abstract: Physics-informed neural networks (PINNs) have shown promising results in solving partial differential equations (PDEs). Nevertheless, for some challenging PDEs, standard PINNs can fail to converge. We propose a novel curriculum learning strategy that addresses this limitation. Our method leverages an extension of the implicit function theorem to guide the training process along the solution manifold of the parameterized differential equation, starting from an easy-to-solve problem and progressively moving towards a hard-to-solve one. We establish a theoretical link between our approach and natural gradient descent, giving rise to a new effective curriculum learning algorithm allowing us to solve difficult PDEs such as Eikonal and Hamilton Jacobi-Bellman equations.

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Submission Number: 42